Matrix multiplication and composition of operators on the direct sum of an infinite sequence of Banach spaces
Published online by Cambridge University Press: 26 October 2001
Let [Efr ] be a Banach space with a normalized, 1-unconditional basis. Each operator on the [Efr ]-direct sum of a sequence ([Xfr ]i)i∈ℕ of Banach spaces corresponds to an infinite matrix. We study whether this correspondence is multiplicative, in which case we say that matrix multiplication works. We prove that matrix multiplication works if at least one of the following two conditions is satisfied:
(i) for each i ∈ ℕ, each operator from [Xfr ]i to [Efr ] is compact;
(ii) the basis of [Efr ] is shrinking and, for each i ∈ ℕ, each operator from [Efr ] to [Xfr ]i is compact.
In the case where [Efr ] is either c0 or [lscr ]p, where 1 [les ] p < ∞, the converse also holds.
- Research Article
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 131 , Issue 1 , July 2001 , pp. 165 - 183
- 2001 Cambridge Philosophical Society